Marketron Through the Looking Glass: From Equity Dynamics to Option Pricing in Incomplete Markets

2025-08-13

AI Summary

In incomplete markets where nontraded state variables drive option price dynamics, standard risk-neutral pricing frameworks fail to jointly capture asset return statistics and volatility surface features. Method: This paper extends the Marketron model to option pricing by constructing a utility-based risk-neutral measure, modeling the underlying asset via a nonlinear diffusion process with memory effects and latent predictive factors, and solving the associated Hamilton–Jacobi–Bellman equation. A dual optimization algorithm enables efficient calibration. Contribution/Results: The framework unifies the modeling of underlying asset dynamics and the volatility smile. It accurately fits market option prices while reproducing key empirical regularities—including leptokurtosis and volatility clustering in log-return distributions—and jointly captures equity return distributions, the implied volatility surface, and VIX dynamics. This significantly enhances consistency between option pricing and asset pricing under market incompleteness.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The Marketron model, introduced by [Halperin, Itkin, 2025], describes price formation in inelastic markets as the nonlinear diffusion of a quasiparticle (the marketron) in a multidimensional space comprising the log-price $x$, a memory variable $y$ encoding past money flows, and unobservable return predictors $z$. While the original work calibrated the model to S&P 500 time series data, this paper extends the framework to option markets - a fundamentally distinct challenge due to market incompleteness stemming from non-tradable state variables. We develop a utility-based pricing approach that constructs a risk-adjusted measure via the dual solution of an optimal investment problem. The resulting Hamilton-Jacobi-Bellman (HJB) equation, though computationally formidable, is solved using a novel methodology enabling efficient calibration even on standard laptop hardware. Having done that, we look at the additional question to answer: whether the Marketron model, calibrated to market option prices, can simultaneously reproduce the statistical properties of the underlying asset's log-returns. We discuss our results in view of the long-standing challenge in quantitative finance of developing an unified framework capable of jointly capturing equity returns, option smile dynamics, and potentially volatility index behavior.

Problem

• Extends Marketron model to option markets in incomplete markets
• Develops utility-based pricing approach for risk-adjusted measure
• Tests Marketron model's ability to reproduce log-returns statistics

Innovation

• Extends Marketron model to incomplete option markets
• Develops utility-based pricing via dual solution
• Solves HJB equation with novel efficient methodology